Abstract:In this paper, we derive explicit formulae for the number of regular even-valent graphs, with fixed minimal embedding genus, in which both the valence parameter and the number of vertices are allowed to vary. Our results extend the explicit formulae of Ercolani--McLaughlin--Pierce (2008) for genus $0$ and of Ercolani--Lega--Tippings (2023) for genus $1$.
More precisely, we obtain explicit counts $\mathscr{N}_g(2\nu,j)$ -- with $\nu$ and $j$ as variables -- of graphs with $j$ vertices of uniform valence $2\nu$ and minimal embedding genus $g$, for $2\leq g\leq 4$. We also obtain the corresponding formulae for the two-legged counts $\mathcal{N}_g(2\nu,j)$. The method applies to $g\geq 5$, with increasing computational effort as $g$ increases. Finally, we derive leading-order large-valence asymptotics for these counts when $g\leq 4$, and formulate a structural conjecture for higher genus.
From: Tomas Lasic Latimer Mr [view email]
[v1]
Fri, 2 May 2025 23:42:42 UTC (233 KB)
[v2]
Thu, 18 Sep 2025 17:04:24 UTC (226 KB)
[v3]
Mon, 15 Jun 2026 11:20:49 UTC (273 KB)
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